Abstract—In this paper the attitude dynamics of a
spacecraft (SC) is considered under the control by the
geometrical relocation of the internal position of the center of
mass relatively the SC-frame. This relocation can, firstly,
change the SC inertia tensor and, secondly, change the lever
of force at the creation of the control torque from the jet
thrust of the SC reaction-propulsion unit; so the SC attitude
dynamics is complex and nonstationary. The internal
relocation of the mass center is realized by moving the
internal mass inside the SC-frame, and then the SC attitude
control can be constructed basing on this mass internal
motion.
In this work the displacements of the internal mass are
formed by the control system proportionally to the
components of the angular velocity of the SC. In this case the
dynamics can have the regimes with non-trivial effects,
including Shilnikov’s attractors.
Index Terms—spacecraft, attitude, control, internal
movable mass, Shilnikov’s attractor
I. INTRODUCTION
his work is dedicated to an investigation of nonlinear
aspects of attitude dynamics of spacecraft (SC) with the
geometrically relocatable internal position of the center of
mass relatively the SC frame. The internal relocation of the
mass center (fig.1) is realized by moving the internal mass
m inside the SC-frame. This relocation can, firstly, change
the SC inertia tensor and, secondly, change the lever of
force at the creation of the control torque from the jet thrust
(P) of the SC reaction-propulsion unit. Basing on this
dynamical effects it is possible to build the control system
for attitude control of SC relatively the inertial frame
OXYZ, where the origin O corresponds to the point
coincided with the actual center of mass of complete system
(the main SC-frame and the internal mass).
Let us consider the motion of the SC with moving
internal mass at its controlled displacements proportionally
to the SC angular velocity components.
Manuscript received December 10, 2016. This work is partially supported
by the Russian Foundation for Basic Research (RFBR#15-08-05934-А), and
by the Ministry of Education and Science of the Russian Federation in the
framework of the State Assignments to higher education institutions and
research organizations (Project# 9.1616.2017/ПЧ).
A. V. Doroshin is with the Space Engineering Department (Division of
Flight Dynamics and Control Systems), Samara National Research University
(SSAU, 34, Moskovskoe Shosse str., Samara, 443086, Russia; e-mail:
[email protected]; [email protected]; phone/fax: +7 846 335-18-63); IAENG
Member No: 110131.
In is well known, the attitude dynamics of SC (rigid
bodies with internal degrees of freedom) can be complex
and nonstationary, and the phase space of the system can
collect the complex irregular dynamical objects, e.g. strange
chaotic attractors, chaotic homo/heteroclinic orbits. These
nonlinear aspects are very interesting from the
mathematical point of view [1-27], and, moreover,
investigated properties of nonlinear phenomena can be used
to design new effective control systems. Therefore,
discovering and studying nonlinear irregular objects is the
main goal of the work.
Fig. 1. The spacecraft with moving internal mass and
corresponding coordinates systems
In the next sections is described the corresponding
mechanical and mathematical models of the motion of SC
with the relocatable internal mass. Also the numerical
modeling results are presented, including the emergence of
well-known Shilnikov’s attractors [1-3] in the phase space.
Attitude Dynamics of Spacecraft with Control
by Relocatable Internal Position of Mass Center
Anton V. Doroshin, Member, IAENG
T
Proceedings of the International MultiConference of Engineers and Computer Scientists 2017 Vol I, IMECS 2017, March 15 - 17, 2017, Hong Kong
ISBN: 978-988-14047-3-2 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
IMECS 2017
II. THE MECHANICAL AND MATHEMATICAL MODELS
Let us consider the attitude dynamics of SC which
consists from the main body-frame (with its own mass
center C and mass M) and internal moving mass m, which
can relocate its position inside the main SC body only along
plane Π (fig.1). The moving mass-point m in technical
sense can correspond to the mass center of internal weights
of a system of special regulators or multifunctional
equipment which has technical/constructional opportunities
to implement the controlled planar displacements relative
the body-frame Cxyz depending on time (x=x(t), y=y(t)).
The mass center of complete system O has the following
values of coordinates in the body-frame:
;O Ox t mx t m M x t y t y t (1)
where m m M . Then it is possible to involve the
main central connected coordinates frame Oξηζ of the
complete mechanical system (fig.1) with the origin O in the
system mass center and with axes parallel to axes of the
body-frame Cxyz; and it is clear, that
1 ; 1 ;O Ox x x y y y z (2)
Assume that the inertia tensor of the main SC body in its
own connected coordinates system Cxyz has the general
diagonal form Ixyz=diag[Ab, Bb, Cb]. Taking into view the
displacements of general axes, the inertia tensor of the SC
body in the coordinates system Oξηζ will have the form
which is depended on time (due to dependencies (1)):
2
2
2 2
0
0
0 0
b O O O
O O b O
b O O
A My Mx y
t Mx y B Mx
C M x y
I I (3)
Then the angular momentum of the SC-frame in connected
coordinates system Oξηζ can be written
body K I ω (4)
where , ,T
p q rω is the vector of the absolute angular
velocity of the SC in projections on the connected
coordinates system Oξηζ.
The angular momentum of the moving mass relative the
point O in projections on the axes Oξηζ can be written as
follows:
2
2
2 2
;
0
0 ;
0 0
0, 0,
e r
m m m
e
m
Tr
m
m m p
m m q
rm
m
K K K
K
K
(5)
where e
mK corresponds to the external part of the motion
(the angular motion of the mass m around the point O as
the part of the “frozen” SC-frame); and r
mK is the relative
angular momentum component of the “unfrozen” point
(relatively the body).
Then the dynamical equations of the system angular
motion follow from the law of the angular momentum
changing, written with the help of the local derivation in
the connected coordinates system Oξηζ:
e
body m body m O
d
dt K K ω K K M (6)
where e
OM is the vector of the external torques.
Let us consider the SC angular motion under the action
only the jet thrust P and small spin-up torque Mz forming
by the SC reaction-propulsion unit (fig.1). Assume that the
modules of the force P and the torque Mz are constant and
quite small (the smallness of these modules allows to
consider the SC as the system with constant mass, without
considering the change of mass of the actuating medium in
the propulsion unit). In this case the vector of external
torques in projections on Oξηζ has the form:
, ,Te
O O O zy P x P M M (7)
Now for constructing the attitude control system it is
needed to define the control laws for the x-y-displacements
of the internal mass-point m. In this work the linear form of
the feedback control system is selected, which provides for
the internal mass-point the following displacements-
dependencies (on p, q, r components):
0
0
;
,
x x x x
p q r
y y y y
p q r
x x t c p t c q t c r t c
y y t c p t c q t c r t c
(8)
where j
ic are constants , , ,0 , ,i p q r j x y .
To describe the angular position of the SC in the inertial
space OXYZ the well-known Euler’s kinematical equations
can be added:
cos sin ;
( sin cos ) sin ;
ctg ( sin cos )
p q
p q
r p q
(9)
So, the dynamical equations (6) together with the control
laws (8) and the links (1), (2), and with kinematical
equations (9) form the complete dynamical system for the
modeling the attitude dynamics of the SC with the feedback
control of the relocatable position of the center of mass.
The substantial simplification of the equations can be
fulfilled at the assumption of the negligibly small relative
angular momentum of the moving internal mass (in
comparison with sum of the external angular momentum of
the mass-point and the angular momentum of the main
body), i.e. .r e
m body m K K K And then it is possible to
move the corresponding terms to the right part of
dynamical equations and to consider it as a small
perturbing torque. Moreover, in this work the following
extreme simplification is taken:
0r
m K (10)
In spite of the simplification (10) the equations (6) in the
scalar form remain cumbersome and implicit, therefore we
do not present the reducing results.
In the next section the corresponding integration results
of the equations (6) with control laws (8) at the
simplification (10) are shown. As it will be numerically
verified, this dynamical system has very interesting
dynamical behavior, and, among other things, it contains
Shilnikov’s attractors.
Proceedings of the International MultiConference of Engineers and Computer Scientists 2017 Vol I, IMECS 2017, March 15 - 17, 2017, Hong Kong
ISBN: 978-988-14047-3-2 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
IMECS 2017
III. MODELING RESULTS
Basing on the obtained in the previous section
mathematical model we can realize numerical experiments
with the integration dynamical equations (6) at different
sets of the system parameters, initial conditions, and
coefficients of the control laws (8).
A. The Case A
Let us to present the integration results (fig.2, fig.3) at
the parameters and coefficients (tabl.A) which quite
correspond to the class of small SC (micro-SC).
At the fig.2-a we can see the complex phase-trajectory
(in the framework of the classical mechanic it is called as
“polhode”), which evolves in the time and proceeds to the
Shilnikov’s attractor depicted separately at the fig.3-a. To
realize this phase trajectory the control system must
generate (basing on the p-q-r-feedback) as the result the
relative motion of the internal moving mass (fig.2-b), that
also evolves to the repeated cycles (fig.3-b) along the
Shilnikov’s attractor (fig.3-a). Also it is possible to indicate
the interesting intermediate spiral section of the attractor.
(a)
(b)
Fig. 2. The polhode/trajectory (a) in the phase space {p, q, r} and the time-
history (b) for coordinates of the moving internal mass
TABLE A
The SC control parameters
INERTIA
MOMENTS
[kg∙m2], MASS
[kg], FORCE [N],
TORQUE [N∙m]
CONTROL
COEFFICIENTS j
ic [m s]
INITIAL ANGULAR
VELOCITIES [1/S]
AND ANGLES
[RAD]
Ab 8 x
pc 0.00025
p0 1.0 Bb 6 x
qc 0.0625
Cb 4 x
rc -0.0375
M 60 0
xc 0
q0 0.0 m 6 y
pc 0.075
P 1 y
qc 0.00025
Mz 1 y
rc 0.0375 r0 0.0
μ 0.09 0
yc 0
(a)
(b)
Fig. 3. The Shilnikov’s attractor (a) in the phase space {p, q, r} and the
time-history (b) for coordinates of the moving internal mass
Proceedings of the International MultiConference of Engineers and Computer Scientists 2017 Vol I, IMECS 2017, March 15 - 17, 2017, Hong Kong
ISBN: 978-988-14047-3-2 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
IMECS 2017
B. The Case B
The second case of the Shilnikov’s attractor (fig.4) can
be initiated in the phase space at the parameters (tabl.B).
TABLE B
The SC control parameters
INERTIA
MOMENTS
[kg∙m2], MASS
[kg], FORCE [N],
TORQUE [N∙m]
CONTROL
COEFFICIENTS j
ic [m s]
INITIAL ANGULAR
VELOCITIES [1/S]
AND ANGLES
[RAD]
Ab 8 x
pc 0.0375
p0 1.0 Bb 6 x
qc 0.75
Cb 4 x
rc -0.375
M 60 0
xc -0.0373
q0 0.0 m 6 y
pc 0.75
P 1 y
qc 0.0375
Mz 1 y
rc 0.375 r0 0.0
μ 0.09 0
yc -0.75
(a)
(b)
Fig. 4. The Shilnikov’s attractor (a) in the phase space {p, q, r} and the
time-history (b) for coordinates of the moving internal mass
C. The Case C
The additional case (tabl.C) of the dynamics with the
Shilnikov’s attractor (fig.5) is possible.
TABLE C
The SC control parameters
INERTIA
MOMENTS
[kg∙m2], MASS
[kg], FORCE [N],
TORQUE [N∙m]
CONTROL
COEFFICIENTS j
ic [m s]
INITIAL ANGULAR
VELOCITIES [1/S]
AND ANGLES
[RAD]
Ab 1.5 x
pc 0.0005
p0 0.2 Bb 1.2 x
qc 0.125
Cb 1 x
rc -0.075
M 10 0
xc 0
q0 0.3 m 1 y
pc 0.15
P 1 y
qc 0.0005
Mz 0.1 y
rc 0.075 r0 0.6
μ 0.09 0
yc 0
(a)
(b)
Fig. 5. The Shilnikov’s attractor (a) in the phase space {p, q, r} and the
time-history (b) for coordinates of the moving internal mass
Proceedings of the International MultiConference of Engineers and Computer Scientists 2017 Vol I, IMECS 2017, March 15 - 17, 2017, Hong Kong
ISBN: 978-988-14047-3-2 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
IMECS 2017
Also the strong dissipative regimes can realize (fig.6).
Fig. 6. The dissipative possible regime
So, the modeling results clearly show possibility of using
the attitude control by the displacement of the mass center
due to moving the internal mass, even in the cases of the
intentional creating complex regimes with Shilnikov’s
attractors in SC dynamics.
IV. CONCLUSION
In the paper the SC attitude dynamics was considered
under the control by the geometrical relocation of the
internal position of the center of mass. This relocation
fulfilled with the help of moving internal mass.
Undoubtedly, the considered control method uses the
well-known dynamical scheme, which based on the
dynamics of rigid bodies with internal degrees of freedom
(moving parts), but the suggested way of SC attitude control
implementation can be indicated as original. Concretized
constructional parameters, including mass-inertia and
geometrical parameters, and real intervals of dynamical
parameters, certainly, must be defined/predefined in the
separate self-contained research.
The dynamics of the SC is complex, and the Shilnikov’s
attractors can realize. This nonlinear dynamical object in its
turn can generate the Smale’s horseshoes and produce the
corresponding dynamical chaos. The extended study of the
SC with attitude control by mowing internal mass and also
chaotic aspects investigation in the SC motion along the
Shilnikov’s attractors are the next tasks for further
research.
ACKNOWLEDGMENTS
This work is partially supported by the Russian
Foundation for Basic Research (RFBR#15-08-05934-A),
and by the Ministry of education and science of the Russian
Federation in the framework of the State Assignments
program (№ 9.1616.2017/ПЧ).
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Proceedings of the International MultiConference of Engineers and Computer Scientists 2017 Vol I, IMECS 2017, March 15 - 17, 2017, Hong Kong
ISBN: 978-988-14047-3-2 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
IMECS 2017